Symbolic examples of the AnalyticalApproximations`LdeApprox` package

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This notebook illustrates AnalyticalApproximations`LdeApprox` package capability of doing symbolic polynomial approximation of an LDE solution.

This loads the package.

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Example 1

Simple boundary value problem.

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Finding polynomial approximation for solution of the BVP.

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Using Mathematica function DSolve to get exact solution of the BVP.

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Comparing exact and approximate results using Mathematica function Plot3D.

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Comparing exact and approximate results for symbolic_10.gif

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Example 2

Initial value problem.

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Finding polynomial approximation for solution of the IVP.

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Using Mathematica function DSolve to get exact solution of the IVP.

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Comparing exact and approximate results using Mathematica function Plot3D.

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Example 3

Boundary value problem.

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Finding polynomial approximation for solution of the BVP.

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Using Mathematica function DSolve to get exact solution of the BVP.

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Comparing exact and approximate results for symbolic_27.gif

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Example 4

Boundary value problem.

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Finding polynomial approximation for solution of the IVP.

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Exact solution of the IVP.

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Comparing exact and approximate results using Mathematica function Plot3D.

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Comparing exact and approximate results for symbolic_38.gif

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Example 5

Boundary value problem.

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Finding polynomial approximation for solution of the BVP.

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Using Mathematica function DSolve to get exact solution of the BVP.

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Comparing exact and approximate results using Mathematica function Plot3D.

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Comparing exact and approximate results for symbolic_49.gif

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Example 6

Initial value problem.

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Finding polynomial approximation for solution of the BVP.

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Unfortunately DSolve can not find exact solution of the IVP.

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Nevertheless the exact solution is as follows.

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Comparing exact and approximate results using Mathematica function Plot3D.

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Example 7

Initial value problem.

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Finding polynomial approximation for solution of the IVP.

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Using Mathematica function DSolve to get exact solution of the IVP.

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Comparing exact and approximate results for λ=1/2*(2*m+3) using Mathematica function Plot3D.

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Comparing exact and approximate results using Mathematica function Plot3D.

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In fact for λ= (2 m+3), λ= 2 (2 m+3), λ= 3 (2 m+3) the exact and approximate solutions are the same.

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